Prelims
a) R9C5 = 3
b) R23C1 = {18/27/36/45}, no 9
c) R23C2 = {59/68}
d) R23C8 = {39/48/57}, no 1,2,6
e) R23C9 = {29/38/47/56}, no 1
f) R67C4 = {69/78}
g) R67C6 = {15/24}
h) R9C34 = {18/27/36/45}, no 9
i) R9C67 = {69/78}
j) 20(3) cage at R5C8 = {389/479/569/578}, no 1,2
k) 12(4) cage at R1C1 = {1236/1245}, no 7,8,9
l) 12(4) cage at R6C7 = {1236/1245}, no 7,8,9
Steps resulting from Prelims
1a. R9C5 = 3, clean-up: no 6 in R9C34
1b. 12(4) cage at R1C1 = {1236/1245}, 1,2 locked for R1
1c. 1 in N3 only in R23C7, locked for C7 and 24(6) cage at R2C7, no 1 in R345C6
1d. 12(4) cage at R6C7 = {1236/1245}, 1 locked for C8
2. 45 rule on R1 1 innie R1C5 = 8
2a. 18(3) cage at R4C5 = {279/459/567}, no 1
3. 45 rule on C1234 2 innies R28C4 = 12 = [39/48/57/75}, no 1,2,6, no 9 in R2C4, no 4 in R8C4
4. 45 rule on C6789 2 innies R28C6 = 9 = [18/27/36/72] (cannot be {45} which clashes with R67C6), no 4,5,9, no 6 in R2C6, no 1 in R8C6
5. 45 rule on R6789 3 innies R6C159 = 20 = {389/479/569/578}, no 1,2
6. 45 rule on N3 1 outie R1C6 = 2 innies R23C7 + 3
6a. Min R23C7 = 3 -> min R1C6 = 6
6b. R1C6 = {679} -> R23C7 = 3,4,6 contains 1 = {12/13/15}, no 4,6,7,8,9
7. 45 rule on N9 1 innie R9C7 = 2 outies R6C78 + 3
7a. Max R6C78 = 6, no 6 in R6C78, no 5 in R6C8
8. 45 rule on N6 3 innies R5C7 + R6C78 = 9 = {126/135/234}, no 7,8,9
9. 45 rule on N7 2 outies R6C23 = 1 innie R9C3 + 3, IOU no 3 in R6C2
10. 45 rule on N8 4 innies R79C46 = 20
10a. Min R7C4 + R9C6 = 13 -> max R7C6 + R9C4 = 7, no 7,8 in R9C4, clean-up: no 1,2 in R9C3
11. Hidden killer quad 6,7,8,9 in R1C6, R28C6, R345C6 and R9C6 for C6, R19C6 = {6789}, R28C6 contains one of 6,7,8 -> R345C6 must contain one of 6,7,8,9
11a. 24(6) cage at R2C6 = {123459/123468/123567}
11b. 7,8 of {123468/123567} must be in R345C6 -> no 6 in R345C6 (because they can only contain one of 6,7,8)
12. 2 in N3 only in R23C7 = {12} => R1C6 = 6 (step 6) => 9 in R1 only in R1C789 or 2 in R23C9 = {29} -> no 9 in R23C8 (implied locking-out cages), clean-up: no 3 in R23C8
12a. R23C9 = {29/38/56} (cannot be {47} which clashes with R23C8), no 4,7
13. 25(4) cage at R1C6 contains 7,9 = {3679} (only remaining combination, cannot be {4579} which clashes with R23C8), 3,6 locked for R1, 3 also locked for N3, clean-up: no 8 in R23C9
14. R23C8 = {48} (hidden pair in N3), locked for C8
14a. 7 in N3 only in R1C789, locked for R1
15. R23C1 = {18/27/36} (cannot be {45} which clashes with 12(2) cage at R1C1, ALS block), no 4,5
16. 45 rule on N1 2 innies R23C3 = 1 outie R1C4 + 10
16a. R1C4 = {1245} -> R23C3 = 11,12,14,15 = {38/39/68/78} (other combinations for 11,12,14 clash with the remaining values for 12(4) cage at R1C1 while R23C3 = {69} clashes with R23C2), no 1,2,4,5
16b. Killer pair 8,9 in R23C2 and R23C3, locked for N1, clean-up: no 1 in R23C1
16b. 1,4 in N1 only in R1C123, locked for R1
16c. R1C4 = {25} -> R23C3 =12,15 = {39/78}, no 6
17. 45 rule on N2 4 innies R13C46 = 22 = {1579/2479/2569/4567} (cannot be {3469} because R1C4 only contains 2,5), no 3
17a. 24(6) cage at R2C6 = {123459/123468/123567}, CPE no 3 in R5C4
18. 45 rule on N9 3 innies R7C78 + R9C7 = 15 = {159/168/249/258/267/456} (cannot be {348/357} because 12(4) cage at R6C7 cannot contain both of 3,4 or both of 3,5), no 3
19. 45 rule on N69 2 innies R59C7 = 12 = [39/48/57], no 2,6 in R5C7, no 6 in R9C7, clean-up: no 9 in R9C6
19a. R5C7 + R6C78 (step 8) = {135/234}, 3 locked for N6
19b. Killer pair 4,5 in R5C7 + R6C78 and 20(3) cage at R5C8, locked for N6
20. 20(3) cage at R5C8 = {479/569/578}
20a. 6 of {569} must be in R56C9 (R56C9 cannot be {56/59} which clash with R23C9), no 6 in R5C8
20b. 16(3) cage at R4C7 = {169/178/268} (cannot be {259} which clashes with 20(3) cage)
21. R28C4 = 12 (step 3), 15(2) cage at R67C4 -> R2678C4 = 27 must contain 9, locked for C4
22. 45 rule on N47 2 innies R59C3 = 9 = [18/27/45/54], R5C3 = {1245}
23. 45 rule on N47 1 innie R5C3 = 1 outie R9C4 -> whichever value is in R9C4 must also be in 28(6) cage at R2C3 in R5C3
23a. 1 in C4 only in R345C4 + R9C4 -> 28(6) cage must contain1
23b. 28(6) cage = {123679/124678/134569/134578} (cannot be {123589/124579} which clash with R1C4 using “clone” R5C3 = R9C4)
24. R23C3 (step 16c) = {39/78}
24a. 28(6) cage at R2C3 = {123679/124678/134569} (cannot be {134578} which clashes with R28C4 using “clone” R5C3 = R9C4), 6 locked for C4, clean-up: no 9 in R67C4
25. Naked pair {78} in R67C4, locked for C4, clean-up: no 4,5 in R28C4 (step 3)
25a. R28C4 = [39], clean-up: no 6 in R3C1, no 9 in R3C3 (step 16c), no 6 in R8C6 (step 4)
26. 28(6) cage at R2C3 (step 24a) = {124678/134569} (cannot be {123679} because 3,7,9 only in R23C3)
26a. 4 of {124678} must be in R5C3 (R59C3 cannot be [18/27] which clash with {78} in R23C3), no 2 in R5C3, clean-up: no 7 in R9C3 (step 22), no 2 in R9C4
27. Killer pair 1,2 in R28C6 and R67C6, locked for C6
27a. 24(6) cage at R2C6 (step 17a) = {123459/123468/123567} -> R23C7 = {12}, locked for C7 and N3, clean-up: no 9 in R23C9
27b. Naked pair {56} in R23C9, locked for C9 and N3
28. R1C6 = 6 (hidden single in R1), clean-up: no 9 in R9C7
28a. Naked pair {78} in R9C67, locked for R9, clean-up: no 1 in R5C3 (step 22), no 1 in R9C4
28b. Naked pair {45} in R9C45, locked for R9
28c. Naked pair {45} in R59C3, locked for C3
29. Naked pair {78} in R7C4 + R9C6, locked for N8 -> R8C6 = 2, R2C6 = 7 (step 4), R9C67 = [87], R67C4 = [87], clean-up: no 2 in R3C1, no 8 in R3C3 (step 16c), no 4 in R67C6
29a. Naked pair {15} in R67C6, locked for C6
30. Naked triple {349} in R345C6, locked for 24(6) cage at R2C7 -> R5C7 = 5, R5C3 = 4, R9C34 = [54]
31. 22(4) cage at R7C5 contains 2,9 = {2569} (only remaining combination), 5,6 locked for C5 and N8 -> R67C6 = [51]
32. 20(3) cage at R5C8 = {479} (only remaining combination) -> R6C9 = 4, R5C89 = {79}, locked for R5 and N6 -> R5C5 = 2
33. R5C5 = 2 -> R46C5 = 16 = {79}, locked for C5 and N5 -> R345C6 = [943], clean-up: no 5 in R2C2
34. R6C7 = 3 -> 12(4) cage at R6C7 = {1236} (only remaining combination) -> R6C8 = 1, R7C78 = [62], R78C5 = [56], R9C8 = 9, R8C78 = 9 = [45], R9C9 = 1, R5C89 = [79], R1C789 = [937], R4C789 = [862], R45C4 = [16]
35. R5C12 = {18} = 9 -> R6C1 = 9, R46C5 = [97]
36. 9 in N7 only in 20(4) cage at R6C2 = {2369} (only remaining combination) -> R7C23 = {39}, locked for R7 and N7 -> R78C9 = [83]
37. R7C1 = 4 -> R89C1 = 9 = [72]
and the rest is naked singles.
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